Flow regimes and wall shear rates determination within a ... · An experimental investigation of a...

Flow regimes and wall shear rates determination within a scrapedsurface heat exchanger

Eric Dumont a,b, Francine Fayolle a,*, Jack Legrand b

a Ecole Nationale Sup�erieure dÕIng�enieurs des Techniques Agricoles et Alimentaires, D�epartement G�enie des Proc�ed�es Alimentaires,

Rue de la G�eraudi�ere, BP 82225, 44322 Nantes Cedex 3, Franceb Laboratoire de G�enie des Proc�ed�es, UPRES EA 1152, Universit�e de Nantes, CRTT, BP 406, 44602 Saint Nazaire Cedex, France

Received 30 July 1999; accepted 6 March 2000

Abstract

An experimental investigation of a scraped surface heat exchanger (SSHE) was undertaken using visual observations and the

electrochemical technique in order to, ®rstly, study the transition between laminar and vortex ¯ows and, secondly, evaluate the wall

shear rates. Visual observations and the electrochemical technique had undergone preliminary testing in a SSHE without blades, a

well-known reference annulus. Then, visual observations and wall shear rates was compared in both geometries. It was established

that ¯ow patterns in a SSHE are noticeably di�erent from those observed in an annular space in the same conditions. In a SSHE, the

formation of the vortices is thwarted by the rotation of the blades (and by the clips of the blades) and the transition between laminar

and vortex ¯ows occurs at Tagc � 80 (Tagc � 45 in the annular space without blades). Local measurements of the shear rate at the

tube wall of the SSHE showed that it is fully controlled by the rotation of the blades. Vortices have a negligible in¯uence in

comparison with that of blade scrapings. Wall shear rates due to blade scraping can reach up to 40 000 sÿ1. Under these conditions,

the lowest value of the clearance between the edge of the blades and the stator is about 50 ´ 10ÿ6 m. Ó 2000 Elsevier Science Ltd. All

rights reserved.

Keywords: Scraped surface heat exchanger; Flow patterns; Vortex ¯ow; Microelectrode; Visual analysis; Wall shear rates

1. Introduction

Scraped surface heat exchangers (SSHE), mainly usedin the food industry, allow highly viscous ¯uids withcomplex rheology (cream cheese, fruit concentrate, icecream, among others) to be processed. The ®rst advan-tage of these exchangers lies in the rotation of a shaftequipped with blades which periodically scrape the ex-change surface in order to prevent crust formation andto promote heat transfer (Fig. 1). The second one lies inthe geometry of the inlet and outlet bowls. In this type ofexchanger, the ¯ow is the result of the superposition of aPoiseuille ¯ow in an annular space and a Couette ¯ow,on which perturbations created by the blades are added.This ¯ow pattern is particularly complex and has onlybeen super®cially studied directly (Trommelen & Beek,1971a; Naimi, 1989; Burmester, Winch & Russel, 1996).Most of the works on the subject tend to model the

geometry of the SSHE as an annular space withoutblades, where the hydrodynamics are well known(H�arr�od, 1986; Abichandani, Sarma & Heldman, 1987).Depending on the rotational speed of the rotor, twodi�erent ¯ow regimes can be shown: laminar or vortex¯ow. Without blades, the change between these tworegimes is due to the appearance of vortices at a criticalvalue of generalised Taylor number (Tagc) varying withthe radius ratio (Rr/Rs; Esser & Grossmann, 1996), in-creasing with the axial ¯ow rate (Reaxg; DiPrima, 1960)and depending on the rheological properties of theproduct used (Wronski & Jastrzebski, 1990). From thevast literature on non-Newtonian annular ¯ow (Giord-ano, Giordano, Prazeres & Cooney, 1998), it can beexpected that plug ¯ow conditions may occur in a SSHEwhen the ¯ow is vortical but close to the laminar ¯owregime (H�arr�od, 1990a). H�arr�od (1990b) experimentallyveri®ed that the most favourable ¯ow pattern in a SSHEoccurs at these conditions. H�arr�od (1990a) also indicatesthat the most favourable ¯ow pattern is obtained by aslight increase of the rotational speed. Thus, it is im-portant to distinguish between laminar and vortex ¯ow

Journal of Food Engineering 45 (2000) 195±207

www.elsevier.com/locate/jfoodeng

* Corresponding author. Tel.: +33-2-51-78-54-80; fax: +33-2-51-78-

54-67.

E-mail address: [email protected] (F. Fayolle).

0260-8774/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.

PII: S 0 2 6 0 - 8 7 7 4 ( 0 0 ) 0 0 0 5 6 - X

in industrial applications in order to promote e�cientmixing and to optimise the rotational speed. Unfortu-nately, the mechanical fragility of treated productssometimes induces working constraints which do notpermit a su�cient rotational speed to allow vortex ¯owand correct mixing. In this case, some of the productstreated in the exchangers show high temperature heter-ogeneities. Industrially, heterogeneities can reach up to30°C (H�arr�od, 1990c). For this author, the weak heattransformation of the product is due to both ¯ow regimeand shape of the shaft and blades. Temperature mea-

surements in both inlet and outlet bowls, which cha-racterise radial and axial mixing in the exchanger, seemsthe best way to emphasise the transition between thedi�erent ¯ow patterns in an industrial SSHE (Miyauchi& Vermulen, 1963; H�arr�od, 1990a). A previous paper(Dumont, Della Valle, Fayolle & Legrand, 2000a) dealswith the thermal evolution of model ¯uids, Newtonianand non-Newtonian, in heating or cooling conditionswithin an industrial SSHE, and allows us to link thephenomena of appearance and disappearance of tem-perature heterogeneities with the changes in the ¯owpattern within the exchanger. Based on literature datadedicated to SSHEs as well as to annular spaces withoutblades (Dumont et al., 2000a), we show that thermallyhomogeneous products can be obtained when Taylorvortices appear in the exchanger. Studies done on theexchanger with and without blades show that the ther-mal behaviour is basically the same for both geometriesbut with a di�erence in critical generalised Taylornumbers value for vortex appearance. The presence ofthe blades should promote the appearance of instabili-ties at lower values of the generalised Taylor number(Tagc � 10 with blades; Tagc � 39 without blades;industrial SSHE).

The aims of this paper are, ®rstly, to observe byvisualisation the transition between laminar and vortex¯ows in order to verify the hypothesis that presence ofblades should promote the appearance of instabilities,and secondly, to determine by means of an electro-chemical technique the wall shear rates generated by therotation of the blades. A preliminary study is carried out

Nomenclature

C0 bulk concentration (mol mÿ3)

D di�usion coe�cient (m2 sÿ1)

d diameter of the microelectrode (m)

dh hydraulic diameter (m); dh � �ds ÿ dr�dr rotor diameter (m)

ds stator diameter (m)

e gap (m)

F Faraday constant (A s molÿ1)

I limiting di�usion current (A)

K consistency coe�cient of the product

(Ostwald law; Pa sn)

l characteristic dimension of the probe along

the direction of the velocity ¯uctuations (m)

L stator length (m)

n ¯ow behavior index of the product

(Ostwald law; dimensionless)

N rotational speed (rpmÿ1)

nL number of blades (dimensionless)

Q ¯ow rate (m3 sÿ1)

Reax � qUddh

l axial Reynolds number (dimensionless)

Reaxg � qU2ÿnd

dnh

K generalised axial Reynolds number

(dimensionless)

Rr rotor radius (m)

Rs stator radius (m)�S average wall shear rate in an annulus (sÿ1)

S local wall shear rate (sÿ1)

Sc ``corrected'' wall shear rate (sÿ1)

Sm wall shear rate at the in¯ow boundary in an

annulus (sÿ1)

SM wall shear rate at the out¯ow boundary in

an annulus (sÿ1)

Smax maximal value of the shear rate at the tube

wall of the SSHE (sÿ1)

Smin minimum value of the shear rate at the tube

wall of the SSHE (sÿ1)

Sp wall shear rate (sÿ1)

t0 characteristic time of the microelectrode (s)

T temperature (°C)

Ta ��RsÿRr

Rr

qqdh

2

�XRr�l Taylor number (dimensionless)

Tag ��RsÿRr

Rr

qqdn

h

2n�XRr�2ÿn

K generalised Taylor number (dimensionless)

Tac, Tagc critical values of Taylor numbers

(dimensionless)

Ud mean axial velocity in the annular space

(m sÿ1)

z number of electrons (dimensionless)

_c shear rate (sÿ1)

d clearance between the edge of the blade and

stator (m)

l dynamic viscosity for Newtonian ¯uid

(Pa s)

q density (kg mÿ3)

X rotating speed of the rotor (rad sÿ1)

Fig. 1. Principal design of SSHE (by courtesy of Duprat company).

196 E. Dumont et al. / Journal of Food Engineering 45 (2000) 195±207

in the SSHE without blades (i.e., an annular space whichrepresents a well-known reference geometry) in order tovalidate results given by visual observations and theelectrochemical method.

2. Materials and method

2.1. Scaled-down model of SSHE

The work was carried out on a scaled-down model ofthe Duprat TR 13 ´ 60 industrial exchanger. All di-mensions are given in Table 1. The scaled-down modelwas used to ful®l visual analyses in order to understandthe part played by the rotational speed and the geo-metrical elements of the exchanger (bowls, shaft andblades) in the transition between laminar and vortex¯ows. But, this scaled-down model had also to allowextrapolation of the results obtained for the model up tothe industrial scale. Thus, the scaled-down was built inorder to present the same thermal behaviour as the in-dustrial SSHE (with variations of the temperature het-erogeneities; Dumont, Fayolle & Legrand, 1999). Thedesign principle used was the retain of the Tag order ofmagnitude found in the industrial exchanger�0 < Tag < 100� when keeping rotational speed at in-dustrial values �0 < N < 10 sÿ1�. The shaft and statordiameters were chosen to be half the ones of the in-dustrial apparatus (which keeps a constant dr/ds ratio).The exchanger volume was calculated taking into ac-count the axial ¯owing velocity which was chosen, for agiven ¯owrate, in order to keep the same residence timeas in the industrial apparatus. The stator length deter-mined in this way (Table 1) represented a compromise athalf scale of two di�erent lengths commonly used in thefood industry �L=ds � 2:3±7:7�. Scaled-down model ofthe SSHE, shown in Fig. 2, allowed the handling ofthermal studies (stator and insulating jacket in stainlesssteel), visualisations and wall shear rates studies (statorof transparent altuglass).

Visualisations and wall shear rates studies were car-ried out under isothermal conditions �T � 25°C�. The

pilot plant included: an inlet tank, a pump, the scaled-down model of the exchanger, a Promag (EndressHauser) electromagnetic ¯owmeter and an experimentalapparatus for electrochemical technique.

2.2. Electrochemical method

Wall shear rates were determined with an electricalmethod. The principle consists in measuring the elec-trical current I delivered by a platinum microelectrodewhich results from a very fast redox reaction:Fe�CN6�3ÿ � eÿ $ Fe�CN6�4ÿ with a large excess ofsupporting electrolyte K2SO4 so as to eliminate themigration current due to the electric ®eld. A voltagedi�erence DV is set between the probe (the workingelectrode) and a large counter electrode (the shaft). IfDV is large enough on the microelectrode, the concen-tration of the reacting ion tends to zero and the currentattains a limit value I. This limiting current is fullycontrolled by di�usion only. If the length of the micro-electrode is very small in the ¯ow direction, the con-centration boundary layer on the electrode is thin; thus,the ¯ow velocity linearly varies throughout the thicknessof this boundary layer. If the ¯ow ¯uctuations are slowenough, the quasi-steady relation can be applied to re-late the wall shear rate, S, to the limiting di�usion cur-rent, I (Reiss & Hanratty, 1963)

S � 1:477

zF

� �I3

D2C30d5

; �1�

where z is the number of electrons involved in the redoxreaction, F the Faraday constant, d the diameter of thecircular microelectrode, C0 the bulk concentration of theactive ions and D is the di�usion coe�cient of these ionsin solution. The limiting di�usion current was measuredon 12 0.4 mm-diameter circular microelectrodes em-bedded axially on the outer cylinder surface (Fig. 3).

2.3. Characteristics of the electrolytes

Three polymeric solutions were used to carry out theexperiments: one Newtonian ¯uid and two shear-thin-ning ¯uids. The viscosity measurements were performedat 25°C with a Couette rheometer TA Instrument AR1000 with imposed torque. The ¯uids were chosen with

Table 1

Geometrical characteristics of industrial SSHE and its scaled-down

model

Industrial

SSHE (1)

Scaled-down

model (2)

Ratio

(1)/(2)

Stator diameter (ds) 0.13 m 0.065 m 2

Stator length (L) 0.60 m 0.38 m 1.58

Ratio (L/ds) 4.61 5.85 0.79

Rotor diameter (dr) 0.080 m 0.040 m 2

Ratio (dr/ds) 0.615 0.615 1

Gap e � �ds ÿ dr�=2 0.025 m 0.0125 m 2

Exchange surface 0.245 (m2) 7.76 ´ 10ÿ2 (m2) 3.16

Fig. 2. Scaled-down model of the SSHE.

E. Dumont et al. / Journal of Food Engineering 45 (2000) 195±207 197

particular care. Indeed, it was di�cult to make highlyviscous solutions complying with the restraints of theelectrochemical method (dissolution of salts in su�cientquantities, chemically inert ¯uid and model ¯uid not toofar away from a food product).

Emkarox HV 45, from ICI, is a mixture of poly-propylene glycol and polyethylene glycol. The dynamicviscosity of the pure products is equal to 3.7 Pa s at25°C. The electrolyte was composed of a mixture ofpotassium ferricyanide, potassium ferrocyanide andpotassium sulphate as supporting electrolyte. The vis-cosities of the di�erent aqueous solutions are given inTable 2. We also studied a low-viscosity solution,polyethylene glycol (PEG) 35 000. Visual observationswere carried out on these solutions mixed with Kalli-roscope and their Newtonian behaviour was checked.

Aqueous solutions of carboxymethylcellulose (CMCfrom Sigma) have a pseudoplastic behaviour. The non-Newtonian characteristics depend on the CMC con-centration. CMC powder was progressively added inagitated cold water. After dissolution of carboxyme-thylcellulose sodium salt, potassium sulphate and po-tassium ferri- and ferrocyanide were added to theelectrolyte. The rheological behaviour was modelled bytwo di�erent power-laws according to the shear ratedomains: 5±80 and 50±1230 sÿ1. The physical propertiesof CMC electrolyte are given in Table 3.

The preparation of guar gum solutions was similar tothat of the CMC solutions, except that the guar gumdissolution was at 40°C. Two power-law equations wereproposed to model the non-Newtonian behaviour ofguar gum solutions: one for shear rates from 2 to 20 sÿ1

and the second for 10 to 1230 sÿ1. Rheological data issummarised in Table 3. Special attention was given forthe determination of the di�usion coe�cients of ferri-cyanide ions in these polymeric solutions (Legrand,Dumont, Comiti & Fayolle, 2000).

2.4. Operating conditions

Each experiment was done as follows: ¯ow rate�Q � 35 l hÿ1�, rotational speed of the rotor (N variedfrom 30 to 600 rpm) and product temperature�T � 25°C� before the inlet were adjusted to the desiredvalues. For each rotational speed, the limiting di�usioncurrents from the microelectrodes were numerically re-corded over 90 s (laminar ¯ow) or over 150 s (vortex¯ow). The generalised axial Reynolds number (Reaxg)and generalised Taylor number (Tag) were used tocharacterise the ¯ow pattern. Visual observations werecarried out on Newtonian solutions mixed with Kalli-roscope AQ 1000 from Kalliroscope Corporation. Kal-liroscope ¯uid is a suspension of microscopic crystallineplatelets which are oriented so as to align their largerdimensions parallel to the streamlines. In the presence ofincident light, areas of varying orientation re¯ect dif-fering light intensities and their variation and movementproduce striking visual images of the streamlines.

3. Results and discussion

3.1. Flow patterns

3.1.1. Without bladesThe ¯ow stability investigations consist of determin-

ing the dependence of the limiting current value I on theangular velocity X of the shaft. At the beginning, thevalue of I is constant with the time (Fig. 4), which isinterpreted as ¯ow without vortices. When a certain

Fig. 3. Schematic representation of a microelectrode embedded on

outer cylinder.

Table 2

Physical properties of Emkarox HV45 solutions at 25°C

Solution Water

(w/w, %)

K3[Fe(CN)6]

(mol mÿ3)

K4[Fe(CN)6]

(mol mÿ3)

K2SO4

(mol mÿ3)

q(kg mÿ3)

l(Pa s)

D ´ 1010

(m2 sÿ1)

HV45-80 20 5.327 5.327 106.5 1080 1.18 0.51

HV45-75 25 5.304 5.304 127.3 1079 0.84 0.70

HV45-70 30 5.283 5.283 158.5 1077 0.60 0.82

HV45-65 35 5.262 5.262 210.5 1073 0.43 1.00

HV45-60 40 5.241 5.241 209.6 1071 0.32 1.15

HV45-55 45 5.220 5.220 229.7 1070 0.23 1.35

PEG ± 5.000 5.000 250.0 1058 0.04 3.50


value of rotational speed is reached, a sudden transfor-mation in I appears. The limiting current value presentsa periodic form (Fig. 4) attributed to the onset of theinstability �Tag � 45�. The maximum value of I corre-sponds to the separation line of two vortices of the samepair at the surface of the stator (maximum value of wallshear rate SM; Fig. 5) while the minimum value corre-sponds to the junction between two pairs of vortices (inthat case, the wall shear rate is minimum, Sm, at thesurface of the stator; Fig. 5). This interpretation iscon®rmed by visual experiments carried out with HV45Emkarox solutions. It is found that the change in thelimiting current evolution corresponds well to the ap-pearance of Taylor vortices (Fig. 6). In this ¯ow, to-roidal vortices encircle the inner cylinder and movethrough the annulus without being disturbed by theaxial ¯ow. The results obtained for very small values ofaxial ¯ow velocity Ud can be applied in stability analysisof Couette ¯ow �Ud � 0�. When Tag exceeds about 240,the limiting current value shows a double periodicvariation (Fig. 4) which indicates the generation of anew vortex ¯ow characterised by wavy oscillations.

These oscillations are generated by a disturbance formedinside the inlet bowl near the inner cylinder. Figs. 6 and7 allow understanding the phenomenon. Without axial¯ow (Fig. 6), Taylor vortices are always perfectlyformed in the annulus with increase of Tag. But, when

Fig. 5. Schematic representation of Taylor vortices in an annulus

without blades.

Table 3

Physical properties of CMC and guar solutions at 25°C

Solution

weight (%)

q(kg mÿ3)

K3[Fe(CN)6]

(mol mÿ3)

K4[Fe(CN)6]

(mol mÿ3)

K2SO4

(mol mÿ3)

D� 1010

(m2 sÿ1)

Shear rate domain _c (sÿ1)

5±80 50±1234

n K (Pa sn) n K (Pa sn)

CMC 0.5 1046 5.0 5.0 300 7.50 0.85 0.11 0.73 0.16

CMC 0.7 1071 5.0 5.0 500 7.50 0.78 0.36 0.65 0.63

CMC 0.9 1047 5.0 5.0 300 7.50 0.70 1.05 0.55 1.98

CMC 1.0 1047 4.0 4.0 300 7.50 0.66 1.64 0.51 3.02

CMC 1.1 1048 4.0 4.0 300 7.50 0.62 2.49 0.48 4.44

Shear rate domain _c (sÿ1)

2±20 10±1234

Guar 0.7 1046 3.0 3.0 300 6.90 0.65 0.78 0.47 1.31

Guar 0.8 1046 4.0 4.0 300 6.90 0.63 1.21 0.44 2.04

Guar 1.0 1043 5.0 5.0 300 6.90 0.51 3.29 0.36 4.86

Guar 1.2 1047 5.0 5.0 300 6.90 0.44 6.40 0.32 8.91

Fig. 4. Examples of record of limiting di�usion current with respect to time in an annulus without blades for a Newtonian solution

�Q � 35 l hÿ1; Reaxg � 3:1�.


axial ¯ow is added to rotational ¯ow, the change ofshaft diameter at the inlet of the annulus (Fig. 1) gives ahydrodynamic perturbation (Fig. 7). When Tag isgradually raised, visual observations show that the dis-turbance concerns only in the ®rst ®ve centimetres of theannulus until a critical value �Tag � 240� as the distur-bance travels axially towards the outlet and creates adoubly periodic ¯ow.

3.1.2. With 2 bladesIn the scaled-down model of the SSHE, with blades

mounted on the inner rotating cylinder, with axial ¯owand no heat transfer, visual investigations show that

the ¯ow stays laminar over a large range of Tag (up to80). For more important values of Tag, ¯ow structurepresents a vortex pattern (Fig. 8) but vortices arethwarted by the rotation of the blades and three sym-metrical zones appear at the ends of the equipment(Fig. 9):· zone a (®rst clip of the blades on the shaft): there are

no vortices in the ¯ow pattern,· zone b: perturbed zone where small vortices are grow-

ing to give birth to Taylor cells. A part of the createdvortices ¯ows in the direction of zone a where it dis-appears; the other part progresses axially in thescaled-down under the action of the ¯owrate,

Fig. 6. View of Taylor vortices at the inlet of the annulus without axial ¯ow for a Newtonian solution (no blades; Tag � 180; N � 10 rps;

Q � 0 l hÿ1; Reaxg � 0�. Flow pattern without perturbation.

Fig. 7. Hydrodynamic perturbation located at the inlet of the annulus with axial ¯ow and for a Newtonian solution (no blades; Tag � 180;

N � 10 rps; Q � 35 l hÿ1; Reaxg � 1:7).


· zone c: vortices acquire stability and move through-out the exchanger towards to the outlet where zoneb then zone a reappear.Consequently, visualisations show that the geometry

of bowls and blades deeply in¯uences the ¯ow pattern atthe edges of the equipment. Exploitation of the limitingdi�usion current given by the probes allows completingthese results.

Fig. 10(a) shows an example of the general evolutionof I with respect to time for a non-Newtonian CMCsolution. The evolution of I is the same for every valueof Tag and for the di�erent ¯uids used (Newtoniansolutions, non-Newtonian solutions of CMC and guar).This representation indicates that the limiting di�usioncurrent is fully controlled by the rotation of blades (foreach value of Tag). Fig. 10(b) shows that minimumvalues of the di�usion current in presence of blades aresigni®cantly greater than those in an annulus withoutblades (laminar and vortex ¯ows). The vortex ¯owpattern is renewed at each blade passage, thereby re-sulting in an unsteady vortex ¯ow which by nature isdi�erent from Taylor vortices. Actually, the scraping ofthe microelectrodes is very fast and creates large ¯uc-tuations. Fig. 10(b) shows the predominance of scrap-ing e�ects under vortices e�ects. Sometimes, inparticular situations of buckling up of the blades, theclearance between the edge of the blade and the wallmay become so thin that no electrolyte occurs on the

microelectrode. In this case, a signal interruption isrecorded.

3.1.3. DiscussionComparison of Figs. 6 and 9 shows that noticeably

di�erent ¯ow patterns in SSHE occur from those ob-served in an annular space in the same conditions. Theformation of vortices is thwarted by the blades (and bythe clips of the blades) and upstream motion (from zoneb to zone a, Fig. 5) of the vortices formation in thedirection of the inlet bowl is observed. Furthermore,visual observations show that the transition betweenlaminar and vortex ¯ows in SSHE occurs for a di�erentcritical value of Taylor number �Tagc � 80� than the onedetermined in an annulus �Tagc � 45�. These resultsdi�er from the main conclusion of the literature whichassumes that ¯ow patterns in SSHE and in annulus arefundamentally comparable (H�arr�od, 1986; Abichandaniet al., 1987). Yet, ¯ow patterns visualisations in SSHEare scarce. Trommelen and Beek (1971a) carried out astudy of the ¯ow pattern in SSHE with a glass outerwall. These authors, who visualised Taylor vortices bydispersing small polyethylene beads in glycerol/watersolutions, declared that the transition from Couette ¯owto Taylor vortices could not be observed very accu-rately. Unfortunately, Trommelen and Beek (1971a) didnot provide photographs of Taylor vortices andrestricted their study to the Couette ¯ow regime.

Fig. 9. View of vortex ¯ow at the inlet of the SSHE for a Newtonian solution (2 blades; Tag � 110; N � 5:9 rps; Q � 35 l hÿ1; Reaxg � 1:9). Zone a ±

without vortices; zone b ± birth-place of vortices and zone c ± developped vortices.

Fig. 8. View of vortex ¯ow in SSHE for a Newtonian solution (2 blades; Tag � 113; N � 6 rps; Q � 35 l hÿ1; Reaxg � 1:9).


Nevertheless, in their conclusion, these authorsstrangely established that the ¯ow is either in the Cou-ette ¯ow regime or in the Taylor vortices regime (withdescription of Taylor cells like those observed in anannulus) and that critical Reynolds number of rotationis equal to that for ¯ow in the annulus. More recently,Burmester et al. (1996) carried out visualisation studiesin a SSHE in which the shell was replaced by a Perspextube. A non-Newtonian model ¯uid (0.4% w/w Carbo-pol EDT 2001) with similar rheological properties tothose of food products is used. High-density polyethyl-ene particles (sieved below 100 lm) are added to this

¯uid. It is observed that particles streak followed par-allel streamlines indicating that the rotational ¯ow is inthe laminar regime. No evidence of Taylor vortices canbe seen in the photographs taken, as expected from therange of Tag studied �Tag < 32�. Lastly, Burmester et al.(1996) established an increase in backmixing withincreasing rotor speed in laminar ¯ow.

Trommelen and Beek (1971a) and Burmester et al.(1996) visual results, associated with those of the presentwork, shows that no transition between laminar andvortex ¯ow occurs at low values of Taylor number inSSHE (as suggested in Benezech (1988)). Consequently,

Fig. 10. (a) Time evolution of limiting di�usion current for a non-Newtonian solution of 0.5% CMC in SSHE (2 blades; Reaxg � 0:5; Q � 35 l hÿ1);

(b) comparison of recorded limiting di�usion current with respect to time in an annulus for three di�erent solutions with and without blades

�Q � 35 l hÿ1�; nb ± no blades; 2b ± 2 blades.


the phenomena of appearance and disappearance oftemperature heterogeneities observed in the industrialSSHE for Tag � 10 (Dumont et al., 2000a) cannot beattributed to the change in the ¯ow pattern within theexchanger. The observed hydrodynamic perturbationsand backmixing located at the edge of the apparatus,and generated by the geometry of bowls and shaft andby ¯owrate, could lead to a change of thermal behaviourinside the bowls and could explain the thermal hetero-geneities within the exchanger. It is interesting to notethat the disturbances observed at the edges of the SSHEdi�er from the numerical analyses of Baccar and Abid(1997) which showed that a pair of rotating vorticesoccurred in the edges of the apparatus. Nevertheless,numerical method used by these authors did not inte-grate the entrance e�ects into the imposed boundaryconditions.

3.2. Wall shear rates

3.2.1. Without bladesFor the laminar ¯ow, S is independent of the axial

position of the microelectrode (I � cte and �S � S) andlinearly increases with N according to Eq. (2) obtainedfrom the Couette ¯ow velocity pro®le for a power-law¯uid (n � 1, for Newtonian solutions)

�SN� 4p

nR2=n

r

R2=ns ÿ R2=n

r

� � : �2�

Eq. (2) shows that �S=N ratio is independent of Tag inlaminar regime. At Tag > Tagc, when Taylor vortices aredeveloped, the electrochemical method allows to obtainaxial measurements of wall shear rate continuously onthe body of the Taylor cells. Visual observations showthat the out¯ow boundary (M) and the in¯ow boundary(m) bound the Taylor cells. SM and Sm correspond to the

value of the mean wall shear for, respectively, eachboundary (Fig. 5). The value of S is higher for out¯owthan for in¯ow. The value given by any probe is locatedbetween SM and Sm and the average wall shear rate �S isgiven by �S � �SM � Sm�=2. For a given microelectrode,an example of the dimensionless representation of themean wall shear rate is presented in Fig. 11. This rep-resentation allows one to verify the theoretical value ofthe �S=N ratio in laminar regime (Eq. (2)) and to detectthe transition between laminar and vortex ¯ows, i.e.,when �S=N ratio becomes a Tag function. Fig. 11 showsthat experimental measurements of �S=N ratio in thelaminar ¯ow agree with theoretical value given byEq. (2) for very viscous Newtonian solutions. The sameagreement is obtained for non-Newtonian CMC solu-tions (Fig. 13). But in the case of non-Newtonian guarsolutions, results obtained show that �S=N decreases withTag in the laminar regime though that �S=N ratio orderof magnitude agrees with Eq. (2). Viscoelastic e�ectsshould be considered. Fig. 11 also shows that vorticesappearance moderately increases the mean wall shearrates which attains 200 sÿ1 for the maximal rotationalspeed (�S=N #20 when N � 10 rps).

3.2.2. With 2 bladesIn the general case of scraping without signal inter-

ruption, the very fast large ¯uctuations of the limitingdi�usion current do not allow one to calculate the wallshear rates using Eq. (1). Indeed, instantaneous wallshear rate can be related to the instantaneous limitingcurrent only in the case of quasi-steady ¯ows (Dumont,Fayolle & Legrand, 2000b). The main di�culty in in-terpreting electrodi�usion measurements comes fromthe need to consider the dynamic response of the con-centration boundary layer at the probe surface to ¯ow¯uctuations. Sobol�õk, Wein and Cermak (1987) propose

Fig. 11. Dimensionless representation of wall shear rates with respect to Tag in an annulus without blades for Newtonian solutions �Q � 35 l hÿ1�.( ) ± HV45-80; ( ) ± HV45-75; ( ) ± HV45-70; ( ) ± HV45-65; ( ) ± HV45-60; ( ) ± HV45-55; ( ) ± PEG.


a model that enables calculation of instantaneous wallshear rate from the measuring limiting di�usion currenteven at large ¯ow ¯uctuations (Eq. (3))

Sc�t� � S�t� � 2

3t0

oSot

� �; �3�

where Sc(t) is the value of the wall shear rate correctedwith respect to ``microelectrode inertia'', S(t) the quasi-steady interpretation of the measured current (Eq. (1)),

and t0 is the characteristic time of the microelectrode. Sand Sc have been calculated in scraping situations inorder to obtain the maximal value of wall shear rate(Smax) and to deduce the clearance between the edge ofthe blade and the wall of the stator. However, the cur-rent variations are so fast (high values of dI/dt) thatcorrections are di�cult to calculate accurately. Thus, theactual value of Smax should be included between the wallshear rate S calculated using Eq. (1) and the ``corrected''

Fig. 12. Evolution of Smax/N (calculated both with Eqs. (1) and (3)) with respect to Tag for Newtonian solutions in SSHE (2 blades). Comparison with

��S=N� measured in an annulus without blades. ( ) ± HV45-80; ( ) ± HV45-75; ( ) ± HV45-70; ( ) ± HV45-65; ( ) ± HV45-60; ( ) ± HV45-55;

( ) ± PEG.

Fig. 13. Evolution of Smax/N (calculated both with Eqs. (1) and (3)) with respect to Tag for non-Newtonian CMC solutions in SSHE (2 blades).

Comparison with ��S=N� measured in an annulus without blades. (a) 0.5% CMC; (b) 0.7% CMC; (c) 0.9% CMC; (d) 1.0% CMC; (e) 1.1% CMC.


wall shear rate Sc calculated using Eq. (3). These wallshear rate curves are both compared in Figs. 12 and 13.For comparison, these ®gures also show the evolution ofthe ratio ��S=N� obtained without blades. We had alsocalculated the minimal value of wall shear rate Smin

corresponding to the wall shear rate between twoscrapings. Figs. 14 and 15 show the evolution of Smin/Nwith respect to Tag for Newtonian and non-NewtonianCMC solutions. Figs. 12±15 clearly reveal that wallshear rates in SSHE are noticeably di�erent from thoseobserved in an annulus without blades. Two observa-tions can be mentioned. First, wall shear rates in SSHE(Smax and Smin) are 10±100 higher than those measuredin an annulus under the same conditions. Second, it is

not possible to determine the transition between laminarand vortex ¯ow which agrees with visual observations asin an annulus without blades. It is not in contradictionwith the good running of the electrochemical technique.In the annulus without blades, probes detect vorticesappearance because it leads to a great change of wallshear rates. In that case, wall shear rates analyses agreewith visual observations. In the case of SSHE with twoblades, Fig. 10(a) and (b) clearly show that vorticesappearance represents a minor phenomenon for the wallshear rates point of view; the major phenomenon beingrepresented by the blades scrapings. Thus, wall shearrates analyses (Figs. 12±15) reveal this fact. Vorticesappearance cannot be detected because its in¯uence

Fig. 14. Evolution of Smin/N with respect to Tag for Newtonian solutions in SSHE (2 blades). Comparison with ��S=N� measured in an annulus

without blades. ( ) ± HV45-80; ( ) ± HV45-75; ( ) ± HV45-70; ( ) ± HV45-65; ( ) ± HV45-60; ( ) ± HV45-55; ( ) ± PEG.

Fig. 15. Evolution of Smin/N with respect to Tag for non-Newtonian CMC solutions in SSHE (2 blades). Comparison with ��S=N� measured in an

annulus without blades. (a) 0.5% CMC; (b) 0.7% CMC; (c) 0.9% CMC; (d) 1.0% CMC; (e) 1.1% CMC.


appears negligible in comparison with the one ofscraping. The changes of evolution of �S=N ratio (Smax/Nand Smin/N), which appear for low and di�erent valuesof Tag, seem to be linked to rheological properties ofsolutions. It is possible that phenomenon like viscousheating of the liquid between the edge of the scraperblade and the wall, which should be considerable(Trommelen & Beek, 1971b), can explain these shifts.Speci®c properties of solutions (slip e�ect) also may beconsidered.

3.2.3. DiscussionSome models of the shear rates in SSHE are proposed

in the literature (Table 4) and are compared with ourexperimental data (Figs. 12±15). The model of Trom-melen and Beek (1971b) is speci®cally devoted to thetransition of the local shear rate corresponding to ascraping of the tube wall. Trommelen and Beek (1971b),who developed a formula to estimate the power re-quirements for the blades in a SSHE, supposed that theedge of the blade is rectangular and the velocity pro®lebetween scraper blade and tube wall is linear. From aseries of measurements of power consumption and fromthe theoretical considerations of Penney and Bell (1967),Trommelen and Beek (1971b) concluded that the clear-ance between the edge of the scraper blade and tube wall�d� is of the order of 1� 10ÿ6 m. In this case, the localshear rate _c (Eq. (4), Table 4) can be estimated as severalhundreds of thousands per second. In our case, experi-mental determination of Smax (Figs. 12 and 13) allowedcalculation of d using Eq. (4). The lowest value of ddetermined in this way is 50 ´ 10ÿ6 m. This experimen-tally measured value is more important than the onecalculated by Trommelen and Beek (1971b). Yet, itcould be found that, in the case of deformation of theblades (buckling), d can be locally less than this value.More recently, Toh and Murakami (1982) analyticallycalculated the power requirements for ¯oating blades.

They consider the shape of the blades, the number ofblades, the scraping angle of the blades and the slit of theblades. But the equations obtained by the mechanicalapproach, which are very voluminous and complex, donot allow calculation of the clearance between the edgeof the scraper and the stator. The other models (Leuliet,Maingonnat & Corrieu, 1986; Maingonnat, Leuliet &Benezech, 1987; Naimi, 1989; H�arr�od, 1990a) are moregenerally devoted to an estimation of a global wall shearrate in the exchanger. Thus, the models of Leuliet et al.(1986) and Maingonnat et al. (1987), based on investi-gations of pressure drop, propose the calculation of theaverage shear rate in SSHE (Eqs. (5) and (6), Table 4).Using industrial conditions, these models show that _c isequivalent to the shear rates calculated in a Couettesystem without blades. In the same way, the model ofNaimi (1989) suggests that the average shear rate inSSHE (Eq. (7), Table 4) is equivalent to the shear rate inan annulus because the term Ud is negligible with respectto the term �2pRrN� and the e�ects of the blades are notconsidered. Lastly, the method proposed by H�arr�od(1990a) seems to be the most simple of all (Eq. (8),Table 4) because the average wall shear rate proposed bythis author is estimated as the arithmetic mean of theshear rate at the tube wall and at the shaft for Newto-nian ¯uids. From H�arr�od (1990a), the method can beexpected to be good for a SSHE if the a�ects of axial¯ow, blades and others design factors can be neglected.The average shear rates corresponding to these di�erentmodels (Eqs. (5)±(8), Table 4) do not exceed �100 sÿ1 or�200 sÿ1. Yet, our local measurements, which completethe di�erent global models, show that shear rates can bevery high in scraping situations. These measurementsrepresent a very interesting information especially forthe mechanical fragile food products. Nevertheless, wallshear rate measurements at the shaft must be realised inorder to complete these ®rst results and to con®rm thevalidity of the di�erent global models.

Table 4

Models of shear rates proposed in the literature

Authors Models Remarks

Trommelen and Beek (1971b) _c � 2pRsNd

�4� Estimation of d: 10ÿ6 m

Leuliet et al. (1986) _c � 3:213ÿ � 104 � 1:45nL nÿ0:7115Q� 23:44Qÿ0:03n0:1754N

� �5� Laminar ¯ow

Maingonnat et al. (1987) _c � n1:3nL Q� bN �6� b and n are given by charts

Naimi (1989) _c �2

��U 2

d � �2pRrN�2q

dh

�7� ±

H�arr�od (1990a) _c � d2s � d2

r

d2s ÿ d2

r

�2pN� �8� Only if the e�ects of axial ¯ow,

blades and others design factors can

be neglected


4. Conclusion

Our investigations in a scaled-down model of an in-dustrial exchanger, visual observations and wall shearrates measurements, show that hydrodynamics in aSSHE and in an annulus are not comparable:1. The transition between laminar and vortex ¯ows in

SSHE occurs for a critical value of generalised Taylornumber �Tag � 80� greater than the one determinedin an annulus �Tag � 45� for the same conditions.Furthermore, the formation of vortices is thwartedby the blades and falling motion of the vortices in for-mation in direction of the inlet bowl are observed.The disturbances of the ¯ow patterns at the edgesof the exchangers, created by the geometry of bowlsand shaft, could explain, instead of ¯ow regimechange, the dramatic change of thermal behaviourobserved in industrial SSHE.

2. In contrast to wall shear rates in an annulus, whichdepend on the ¯ow regime, the wall shear rates in aSSHE appears fully controlled by the rotation ofthe blades. For the wall shear rates point of view,the vortices have a negligible in¯uence in comparisonwith the one of blade scrapings. Then, wall shearrates due to blade scrapings can reach up to 40 000sÿ1. Under these conditions, the lowest value of theclearance between the edge of the blades and the sta-tor is about 50� 10ÿ6 m.

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